Proceedings of Meetings on Acoustics

Thermoacoustic energy harvesting--Manuscript Draft--

Manuscript Number:

Full Title: Thermoacoustic energy harvesting

Article Type: ASA Meeting Paper

Corresponding Author: Andrew AventUniversity of BathBath, North Somerset UNITED KINGDOM

Order of Authors: Andrew Avent

Chris Bowen

Abstract: Thermoacoustics have a key role to play in energy harvesting systems; exploiting atemperature gradient to produce powerful acoustic pressure waves. As the namesuggests, Thermoacoustics is a blend of two distinct disciplines; thermodynamics andacoustics. The field encompasses the complex thermo-fluid processes associated withthe compression and rarefaction of a working gas as an acoustic wave propagatesthrough closely stacked plates in the regenerator of a thermoacoustic device; and thephasing and properties of that wave. Key performance parameters and appropriatefigures of merit for thermoacoustic devices are presented with particular emphasisupon the critical temperature gradient required to initiate the acoustic wave and thethermal properties of the key component; namely the 'stack' or 'regenerator'.Mechanisms for coupling a thermoacoustic prime mover with electromagneticharvesters and piezoelectric transducer materials are also presented which offer thepotential to enhance the energy density attained beyond that possible with linearalternators. Numerical modelling strategies are presented which enable parametricsweeps of the geometric and thermal properties, which influence the efficiency, andperformance of the key components of such devices.

Section/Category: Engineering Acoustics

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Volume 23 http://acousticalsociety.org/

169th Meeting of the Acoustical Society of AmericaPittburgh, Pensylvania

18 - 22 May 2015

Engineering Acoustics: Paper 4pEA11

Thermoacoustic energy harvestingAndrew W AventDept. Mechanical Engineering, University of Bath (UK); a.avent@bath.ac.uk

Christopher R BowenDept. Mechanical Engineering, University of Bath (UK),; msscrb@bath.ac.uk

Thermoacoustics have a key role to play in energy harvesting systems; exploiting a temperature gradientto produce powerful acoustic pressure waves. As the name suggests, Thermoacoustics is a blend of twodistinct disciplines; thermodynamics and acoustics. The field encompasses the complex thermo-fluid pro-cesses associated with the compression and rarefaction of a working gas as an acoustic wave propagatesthrough closely stacked plates in the regenerator of a thermoacoustic device; and the phasing and propertiesof that wave. Key performance parameters and appropriate figures of merit for thermoacoustic devices arepresented with particular emphasis upon the critical temperature gradient required to initiate the acousticwave and the thermal properties of the key component; namely the stack or regenerator. Mechanisms forcoupling a thermoacoustic prime mover with electromagnetic harvesters and piezoelectric transducer mate-rials are also presented which offer the potential to enhance the energy density attained beyond that possiblewith linear alternators. Numerical modelling strategies are presented which enable parametric sweeps of thegeometric and thermal properties, which influence the efficiency, and performance of the key componentsof such devices.

Published by the Acoustical Society of America

Cover Page

1 An introduction to thermoacoustics

Thermoacoustics, as the name suggests, is a blend of two distinct disciplines; thermodynamics andacoustics. The field encompasses the thermo-fluid processes associated with the compression and rarefactionof the working gas as an acoustic wave propagates through a set of closely stacked plates in a thermoacousticdevice. Figure 1 (a) shows a schematic of a thermoacoustic energy harvester fig 1 (b) shows the way in whichan elemental parcel of gas oscillating between the plates of a thermoacoustic stack in an acoustic wave. Thefigure shows the parcel being compressed at one extremity of its displacement and rarefied at the other,experiencing as it does so associated changes in enthalpy. As a result of the presence of the temperaturegradient applied to the plates, heat is added to the parcel when it has been compressed and has elevatedenthalpy. At the other end of its displacement heat is removed when it has lowered enthalpy following itsrarefaction. This has the effect of amplifying an existing acoustic wave and is, in fact, capable of initiatingone from the natural perturbations in the gas, given a sufficient temperature gradient.

Thermoacoustic devices have several particularly interesting characteristics which are of interest in thefield of energy harvesting:

(i) Mechanical energy in the form of an acoustic wave is generated directly from a temperature gradientapplied across a closely spaced stack of thin plates.

(ii) The mechanical (acoustic) energy can be used to move a piezoelectric membrane or linear alternatorto produce electrical energy.

(iii) The working fluid, in which the acoustic wave is generated, is generally noble and/or inert eliminatingthe need for harmful, ozone depleting refrigerants.

(iv) The process involves no phase change and therefore is extremely versatile and capable of operatingover a wide range of temperatures; this is in contrast to devices which operate using the vapour com-pression cycle and as such, are governed by the characteristics associated with an ‘application specific’working fluid.

(v) Control systems can be proportional rather than binary. Binary control has inherent inefficienciesowing to overshoot and tolerance around the ideal temperature (compressors cutting in and out); pro-portional control allows the device to be run at the correct power output for a given load.

(vi) There are few moving parts, they are inherently simple devices and therefore offer the possibility ofbeing both reliable and economic to produce.

(vii) They are able to operate in hostile or inaccessible working environments.

Figure 2 is a schematic representation of the way in which the temperature gradient amplifies the acous-tic wave using the ‘bucket chain’ effect. It can be seen that associated with each displacement (in the xdirection) there is an inertial component (dL) which must be overcome, a viscous resistance (rvdx), acompliance (dC) and associated thermal relaxation (rκ/dx).

In energy harvesting, heat is supplied to the heat exchanger at the hot end of the stack and is used to gen-erate a standing acoustic wave, this is in turn used to produce electrical energy using either a linear alternatoror piezoelectric transducer. Since energy density is a function of frequency it is preferable to be able to drivethese devices at high frequencies. The inertia of linear alternators limits their operating frequency, piezo-electric transducers offer an alternative mechanism for converting the acoustic energy to electrical energy;capable of operating at high frequencies as a result of their low inertia (mass) as investigated by Jensen andRaspet [1]. Nouh et al. [2] have explored mechanisms whereby the careful re-introduction of a small amountof mass to the system can be used to dynamically magnify the strain values attained when an acoustic waveis used to flex a piezoelectric transducer.

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Manuscript

Figure 1: Schematic of a thermoacoustic energy harvester (a) and expanded view of plates (b).

Figure 2: The bucket-chain effect.

1.1 Primary figure of merit in energy harvesting - Onset temperature gradient

The critical temperature gradient is the gradient through the length of the stack, in the direction ofacoustic wave propagation, above which the device will function as a prime mover (or engine) and belowwhich it will function as a refrigerator (or heat pump). The critical temperature gradient in the inviscid limitis given by (Swift [5]):

∇Tcritical =ωA |p1|ρm cp |U1|

where ω is the angular frequency of the wave, A is the cross sectional area between the plates, ρm is themean density of the working fluid, cp is the specific heat capacity of the working fluid at constant pressure,|U1| and |p1| are moduli of the 1st order complex volume flow rate and pressure respectively. The equationrepresents the enthalpy changes undergone by an elemental parcel of woking fluid as it is compressed andrarefied through an acoustic cycle.

In an energy harvesting context, a more useful figure of merit is the ‘onset temperature gradient’. This isthe temperature gradient at which the natural perturbations in the gas are amplified sufficiently to overcomethe viscous and thermal attenuation illustrated in figure 2, producing an acoustic wave. Figure 3 shows therelationship between the critical temperature gradient and the onset gradient; minimising the difference inthese gradients results in the lowest onset temperature for a given device.

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The onset temperature gradient is dependent upon only two aditional variables, the position of the stackin the standing wave with respect to the pressure and velocity anti-nodes (x−x0), (viscous attenuation beinggreatest near a velocity anti-node) and the plate spacing (y0) (thermal relaxation becoming the dominant lossat larger plate spacings). The coordinate system for the following calculation is defined in figure 4.

The hydraulic radius rh is defined as the maximum distance a parcel of fluid can be from a boundary fora given geometry, for a parallel plate stack this is given by;

rh =A

Π= y0

where A is the area of the plate and Π is the length of its perimeter, which in the case of a parallel platearrangement yields y0.

Figure 3: Onset and critical temperature gradients for a thermoacoustic engine.

Figure 4: Coordinate system for parallel plate stack.

According to Yu and Jaworski [6], the critical onset temperature gradient is given by;

Θcrit = 2π(Im[−fv]/|1− fv|2) + (γ − 1)Im[−fv]tan2[2π(x− x0)/λ]

Im−(fk − fv)/[1− fv)(1− σ)]tan[2π(x− x0)/λ](1)

where from Swift [5, p. 88], Nikolaus Rott’s dissipation function [4] (viscous, subscript v and thermal,subscript κ) h for a parallel plate stack is given by;

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h =cosh[(1 + i)y/δ]

cosh(1 + i)y0/δ(2)

and its spacial average f by;

f =tanh[(1 + i)y0/δ]

(1 + i)y0/δ(3)

and where λ is the characteristic wavelength of the device, σ is the Prandtl number of the working fluidand γ is its ratio of specific heat capacities, δ is either the thermal or viscous penetration depth as requiredwhich are given by;

δκ =√

2k/ωρcp =√

2κ/ω and δv =√

2µ/ωρ =√

2ν/ω (4)

Yu and Jaworski [6] numerically evaluated equation 1 for a range of plate spacings (expressed as rh/δ)in a parallel plate stack and at several positions in the standing wave. It was found that the lowest onsettemperature occurs at a plate spacing of y0/δ = 1.5 and when the stack was placed at 5λ/32 from a velocityantinode. Using Helium as a working fluid at atmospheric pressure, critical onset temperature differencesbelow 20K were predicted. Nouh et al. [3] have carried out a numerical evaluation of the onset temperatureusing SPICETM software, modelling the processes in the stack using an AC circuit analogy (commonlyused to model thermaocoustic processes) which appear to confirm the results of Yu and Jaworski [6]. It isintended to undertake a similar analysis, adjusted for the particular operating conditions and geometry ofthe proposed test rig described below.

1.2 Thermoacoustic test rig

In an attempt to provide some empirical validation for these numerical results, a test rig is currentlyunder construction which will allow a thermoacoustic stack to be mounted on a lead screw and movedaxially in an acoustic wave by means of a stepper motor. Figure 5 shows a schematic of the test rig andfigure 6 shows the pressure and velocity nodes in a standing wave (green representing areas of high pressureand blue, low pressure). In the initial study it is proposed that an acoustic standing wave be driven by meansof a linear actuator, temperature sensors will be placed at either end of the stack to record the rate at whichenthalpy is pumped by the stack at a range of different positions in the wave and at different hydraulic radii.

It is proposed that once this ‘proof of concept’ rig has been successfully evaluated, that hot and ambientheat exchangers be included at either end of the stack. This will allow the device to function as a prime-mover and equation 1 to be evaluated taking into account non-linear effects which are difficult to evaluatenumerically.

2 Stack design and fabrication

The heart of any standing wave thermoacoustic device is the stack, consequently, the design and fabri-cation of thermoacoustic stacks is an important part of this project. Figure 7 shows a method of accuratelyspacing the plates which has been designed whereby spacer plates of low melting point alloy are accuratelypierced and interleaved with the plate material (in this case a carbon fibre pre-preg) and placed in a mould.During the autoclaving process the excess resin from the pre-preg fills the voids in the spacer plates left fromthe piercings. When cured the assembly is heated and the molten alloy spun out centrifugally, leaving resinspacer posts between the layers as seen in figure 8. The process has been tested using a patternmaker’s sheetwax in place of the alloy and a section of stack successfully produced giving confidence that the procedureis sound and offers a practical way to fabricate thermoacoustic stacks accurately.

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Figure 5: Test rig currently under construction to determine optimum stack position

Figure 6: Pressure and velocity nodes in an ideal standing wave 2π out of phase.

The spacer plates are 250µm in thickness which corresponds to a rh/δv ratio close to unity, near theideal value predicted by the numerical analysis carried out by Yu and Jaworski [6].

A COMSOL model of a concentric stack is shown in figure 9 which evaluates the thermal stresses andpotential subsequent deformation of the plates when subjected to a temperature gradient of 300K. Figure9 shows the deformation of the outer 10 plates only when they are constrained by the support webs as aresult of the disparity in the radial and circumferential strains. Far less deformation occurs in the parallel

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Figure 7: Low melting point alloy spacer plates pierced and interleaved with stack plate material.

Figure 8: Resin pillars act as spacers following removal of the low melting point alloy.

plate stack proposed (shown in figure 10) provided the support structure surrounding the plates is of thesame material so that there is no disparity between the various thermal expansions. Unfortunately thisarrangement does not benefit from the pre-stressing of the plates that the coiling of the foils in the coiledstack, this increases the stiffness required of these thin foils (50µm). The thermal and structural analysisof various stack configurations is a key focus since in attempting to optimise the stack for a particularhydraulic radius for a low onset temperature gradient and blockage ratio (maximising the free passage ofthe wave through the stack) the plates must also exhibit sufficient stiffness and thermal capacity for a giventhickness.

Anisotropy of thermal properties in the plates.

It is hoped that the analyses presented above will underpin a successful stack design; one in which ameasure of anisotropy of the thermal properties can be introduced through the innovative use of materialswhich favour heat transfer into the plates (in the y direction), hindering leakage of heat axially through theplate (in the x direction). Defining the thermal properties of the plates as tensors represented by vectors inCOMSOL will allow parametric sweeps to be made of a range of values. The most promising candidates forsuch materials appear to be fibre composites in which tow orientations can significantly influence the mate-rials’ thermal properties. Other possibilities are being explored such as etched films and nano-structures.

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Figure 9: Finite element analysis of a concentric stack showing deformation of the plates (X2)

Figure 10: Finite element analysis of a parallel plate stack

Summary

This paper has provided an overview of thermoacoustic research being undertaken for energy harvestingapplications. The rationale behind the proposed thermoacoustic test rig has been explained as a way of

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validating the theoretical ’sweet spot’ for positioning the thermoacoustic stack in the wave (as predicted bythe numerical evaluation of the equation for critical onset temperature gradient). Initial efforts will measurethe rate of change in temperature difference across the stack as the result of an applied standing wave andthe research will progress to producing a standing wave by introducing heat exchangers to the apparatus.An innovative method for accurately fabricating parallel plate thermoacoustic stacks has been presented andthe potential of this method in tailoring the thermal properties of the stack. Aspects to be considered are thepotential for introducing a degree of anisotropy to the thermal properties which may lead to limiting leakageof heat along the temperature gradient in the stack and enhancing efficiency. A finite element analysis of thethermal stresses in two stack arrangements were discussed to examine the thermal distortion of the stack andhow it may be mitigated in a real device. Future work will explore the development of a fully functioningthermoacoustic device that is capable of harvesting low-grade waste heat and producing electrical energyusing piezoelectric a transducer.

Acknowledgements

The research leading to these results has received funding from the European Research Council underthe European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 320963on Novel Energy Materials, Engineering Science and Integrated Systems (NEMESIS).

References

[1] Carl Jensen and Richard Raspet. Thermoacoustic power conversion using a piezoelectric transducer.The Journal of the Acoustical Society of America, 128(1):98–103, July 2010.

[2] M. Nouh, O. Aldraihem, and A. Baz. Optimum design of thermoacoustic-piezoelectric systems withdynamic magnifiers. Engineering Optimization, 46(4):543–561, May 2013.

[3] M. Nouh, O. Aldraihem, and A. Baz. Onset of Oscillations in Traveling Wave Thermo-Acoustic-Piezo-electric Harvesters Using Circuit Analogy and SPICE Modeling. Journal of Dynamic Systems, Mea-surement, and Control, 136(6):061005, August 2014.

[4] Nikolaus Rott. Advances in Applied Mechanics Volume 20, volume 20 of Advances in Applied Mechan-ics. Elsevier, 1980.

[5] Gregory W Swift. Thermoacoustics: A unifying perspective for some engines and refrigerators. TheJournal of the Acoustical Society of America, 2003.

[6] Z Yu and A J Jaworski. Optimization of thermoacoustic stacks for low onset temperature engines. InProceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, volume224, pages 329–337, January 2010.

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